## Arithmétique des surfaces cubiques diagonales. (Arithmetic of diagonal cubic surfaces).(French)Zbl 0639.14018

Diophantine approximation and transcendence theory, Semin., Bonn/FRG 1985, Lect. Notes Math. 1290, 1-108 (1987).
Using the Brauer groups of schemes Manin has defined an obstruction to the validity of the Hasse principle for cubic surfaces [and varieties, see Yu. I. Manin, “Cubic forms. Algebra, geometry, arithmetic” (1974); translation from the Russian original edition (1972; Zbl 0255.14002); see also the 2nd enlarged edition (1986)]. The authors develop an algorithm for the computation of that obstruction in the case of diagonal surfaces ax $$3+by$$ $$3+cz$$ $$3+dt$$ $$3=0$$. They discover the following infinite series of counterexamples to the Hasse principle: $$a=1$$, $$b=p$$ 2, $$c=pq$$, $$d=q$$ 2 where $$p\equiv 2 \bmod 9$$ and $$q\equiv 5 \bmod 9$$ are primes. It is computed also the obstruction for all equations with $$0<a,b,c,d<100$$. That gives 245 counterexamples to the Hasse principle. The known before Bremner example with $$a=1$$, $$b=4$$, $$c=10$$, $$d=25$$ belongs to the list and minimizes the product abcd.
The computations confirm the following conjecture: for rational surfaces over number fields Manin’s obstruction is the only obstruction to the validity of the Hasse principle. - In the special case of diagonal cubic surfaces V the conjecture can be given more precisely: if there exists a prime p dividing one and only one of the coefficients a,b,c,d (supposed to be integer and without cubic factor) then $$\prod_{\ell}V({\mathbb{Q}}_{\ell})\neq \emptyset$$ implies that V($${\mathbb{Q}})\neq \emptyset$$.
Reviewer: A.Parshin

### MSC:

 14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties 14G25 Global ground fields in algebraic geometry 11D25 Cubic and quartic Diophantine equations

### Keywords:

counterexamples to the Hasse principle; cubic surfaces

### Citations:

Zbl 0582.14010; Zbl 0277.14014; Zbl 0255.14002